text: The waves of permanent form propagate with a constant phase velocity (or celerity), denoted as c. If the steady wave motion is in the horizontal x-direction, the flow quantities η and u are not separately dependent on x and time t, but are functions of :
  Further the waves are periodic – and because they are also of permanent form – both in horizontal space x and in time t, with wavelength λ and period τ respectively. Note that Φ(x,z,t) itself is not necessary periodic due to the possibility of a constant (linear) drift in x and/or t:
  with φ(x,z,t) – as well as the derivatives ∂Φ/∂t and ∂Φ/∂x – being periodic. Here β is the mean flow velocity below trough level, and γ is related to the hydraulic head as observed in a frame of reference moving with the waves phase velocity c' (so the flow becomes steady in this reference frame).
  In order to apply the Stokes expansion to progressive periodic waves, it is advantageous to describe them through Fourier series as a function of the wave phase θ(x,t):
  assuming waves propagating in the x–direction. Here  is the wavenumber,  is the angular frequency and  is the phase velocity.
  Now, the free surface elevation η(x,t) of a periodic wave can be described as the Fourier series:
  Similarly, the corresponding expression for the velocity potential Φ(x,z,t) is:
  satisfying both the Laplace equation  in the fluid interior, as well as the boundary condition  at the bed .
  For a given value of the wavenumber k, the parameters: A, B (with ), c, β and γ have yet to be determined. They all can be expanded as perturbation series in ε.  provides these values for fifth-order Stokes's wave theory.
  For progressive periodic waves, derivatives with respect to x and t of functions f(θ,z) of θ(x,t) can be expressed as derivatives with respect to θ:
  The important point for non-linear waves – in contrast to linear Airy wave theory – is that the phase velocity c also depends on the wave amplitude a, besides its dependence on wavelength  and mean depth h. Negligence of the dependence of c on wave amplitude results in the appearance of secular terms, in the higher-order contributions to the perturbation-series solution.  already applied the required non-linear correction to the phase speed c in order to prevent secular behaviour. A general approach to do so is now known as the Lindstedt–Poincaré method. Since the wavenumber k is given and thus fixed, the non-linear behaviour of the phase velocity  is brought into account by also expanding the angular frequency ω into a perturbation series:
  Here ω will turn out to be related to the wavenumber k through the linear dispersion relation. However time derivatives, through , now also give contributions – containing ω, ω, etc. – to the governing equations at higher orders in the perturbation series. By tuning ω, ω, etc., secular behaviour can be prevented. For surface gravity waves, it is found that  and the first non-zero contribution to the dispersion relation comes from ω (see e.g. the sub-section "Third-order dispersion relation" above).